# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 18/en

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Warm-up-exercises

### Exercise

Prove the following properties of the hyperbolic sine and the hyperbolic cosine

1. ${\displaystyle \cosh x+\sinh x=e^{x}.}$
2. ${\displaystyle \cosh x-\sinh x=e^{-x}.}$
3. ${\displaystyle (\cosh x)^{2}-(\sinh x)^{2}=1.}$

### Exercise

Show that the hyperbolic sine on ${\displaystyle {}\mathbb {R} }$ is strictly increasing.

### Exercise

Prove that the hyperbolic tangent satisfies the following estimate

${\displaystyle -1\leq \tanh x\leq 1{\text{ for all }}x\in \mathbb {R} .}$

### Exercise

Prove by elementary geometric considerations the Sine theorem, i.e. the statement that in a triangle the equalities

${\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}}$

hold, where ${\displaystyle {}a,b,c}$ are the side lengths of the edges and ${\displaystyle {}\alpha ,\beta ,\gamma }$ are respectively the opposite angles.

### Exercise

Compute the determinants of plane and spatial rotations.

### Exercise

Prove the addition theorems for sine and cosine using the rotation matrices.

### Exercise

We look at a clock with minute and second hands, both moving continuously. Determine a formula which calculates the angular position of the second hand from the angular position of the minute hand (each starting from the 12-clock-position measured in the clockwise direction).

### Exercise

Prove that the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {\sin n}{n^{2}}}}$

converges.

### Exercise

Determine the coefficients up to ${\displaystyle {}z^{6}}$ in the series product ${\displaystyle {}\sum _{n=0}^{\infty }c_{n}z^{n}}$ of the sine series and the cosine series.

The next tasks require the definition of a periodic function.

A function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$
is called periodic with period ${\displaystyle {}L>0}$, if for all ${\displaystyle {}x\in \mathbb {R} }$ the equality
${\displaystyle f(x)=f(x+L)}$
holds.

### Exercise

Let

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$
be a periodic function and
${\displaystyle g\colon \mathbb {R} \longrightarrow \mathbb {R} }$

any function.

a) Prove that the composite function ${\displaystyle {}g\circ f}$ is also periodic.

b) Prove that the composite function ${\displaystyle {}f\circ g}$ does not need to be periodic.

### Exercise

Let

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$
be a continuous periodic function. Prove that ${\displaystyle {}f}$ is bounded.

Hand-in-exercises

### Exercise (3 points)

Prove that in the power series ${\displaystyle {}\sum _{n=0}^{\infty }c_{n}x^{n}}$ of the hyperbolic cosine the coefficients ${\displaystyle {}c_{n}}$ are ${\displaystyle {}0}$ if ${\displaystyle {}n}$ is odd.

### Exercise (6 points)

Prove that the hyperbolic cosine is strictly decreasing on ${\displaystyle {}\mathbb {R} _{\leq 0}}$ and strictly increasing on ${\displaystyle {}\mathbb {R} _{\geq 0}}$.

### Exercise (4 points)

Let

${\displaystyle \varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}}$

be the space rotation by ${\displaystyle {}45}$ degree around the ${\displaystyle {}z}$-axis counterclockwise. How does the matrix describing ${\displaystyle {}\varphi }$ with respect to the basis

${\displaystyle {\begin{pmatrix}1\\2\\4\end{pmatrix}},\,{\begin{pmatrix}3\\3\\-1\end{pmatrix}},\,{\begin{pmatrix}5\\0\\7\end{pmatrix}}}$

look like?

### Exercise (5 points)

${\displaystyle \sin(x+y)=\sin x\cdot \cos y+\cos x\cdot \sin y}$

for the sine using the defining power series.

### Exercise (4 points)

Let

${\displaystyle f_{1},f_{2}\colon \mathbb {R} \longrightarrow \mathbb {R} }$
be periodic functions with periods respectively

${\displaystyle {}L_{1}}$ and ${\displaystyle {}L_{2}}$. The quotient ${\displaystyle {}L_{1}/L_{2}}$ is a rational number. Prove that ${\displaystyle {}f_{1}+f_{2}}$ is also a periodic function.

### Exercise (5 points)

Consider ${\displaystyle {}n}$ complex numbers ${\displaystyle {}z_{1},z_{2},\ldots ,z_{n}}$ lying in the disc ${\displaystyle {}B}$ with center ${\displaystyle {}(0,0)}$ and radius ${\displaystyle {}1}$, that is in ${\displaystyle {}B={\left\{z\in \mathbb {C} \mid \vert {z}\vert \leq 1\right\}}}$. Prove that there exists a point ${\displaystyle {}w\in B}$ such that

${\displaystyle \sum _{i=1}^{n}\vert {z_{i}-w}\vert \geq n.}$